Optimal M-PAM Spread-Spectrum Data Embedding with Precoding

Abstract

We consider M-level pulse amplitude modulation (M-PAM) spread-spectrum (SS) data embedding in transform domain host data. The process of data embedding can be viewed as delivering information through the channel including additive interference from host that is known to the embedder. We first utilize the knowledge of second-order statistics of host to design optimal carrier that maximizes the signal-to-interference-plus-noise ratio at the decoder end. Then, inspired by Tomlinson–Harashima precoding used in communication systems, a symbol-by-symbol precoding scheme is developed for M-PAM SS embedding to alleviate the impact of the interference which is explicitly known to embedder. For any given embedding carrier and host data, we aim at designing precoding algorithm to minimize the receiver bit error rate (BER) with any given host distortion budget, and conversely minimize the distortion at any target BER. Experimental studies demonstrate that the proposed precoded SS embedding approach can significantly improve BER performance over conventional SS embedding schemes.

Appendix

Proof of Proposition 2

With SS embedding signal (11), the output SINR of MF is

$$\begin{aligned} \mathrm {SINR}= & {} \frac{{\mathbb {E}} \{\Vert A b_i \Vert ^2 \}}{{\mathbb {E}} \left\{ \Vert \mathbf {s}^T ((\mathbf {I}_L - c\mathbf {s}\mathbf {s}^T)\mathbf {x}_i + \mathbf {n}) \Vert ^2 \right\} }\end{aligned}$$

(42)

$$\begin{aligned}= & {} \frac{\frac{3}{M^2-1} A^2}{\mathbf {s}^T \left( (\mathbf {I}_L - c\mathbf {s}\mathbf {s}^T) \mathbf {R}_{\mathrm { {x}}}(\mathbf {I}_L - c\mathbf {s}\mathbf {s}^T) + \sigma ^2_n \mathbf {I}\right) \mathbf {s}} \nonumber \\= & {} \frac{ \frac{3}{M^2-1} A^2}{\mathbf {s}^T \mathbf {R}_{\mathrm {{x}}} \mathbf {s} -2c\mathbf {s}^T\mathbf {R}_{\mathrm { {x}}}\mathbf {s} + c^2\mathbf {s}^T\mathbf {R}_{\mathrm { {x}}} \mathbf {s} + \sigma ^2_n} \nonumber \\= & {} \frac{\frac{3}{M^2-1} A^2}{\alpha -2 c \alpha + c^2\alpha + \sigma ^2_n} \end{aligned}$$

(43)

where we define $\alpha \triangleq \mathbf {s}^T \mathbf {R}_{\mathrm { {x}}} \mathbf {s}$. By applying ${\mathcal {D}} =\frac{3}{M^2-1} A^2 + c^2 \mathbf {s}^T \mathbf {R}_{\mathrm { {x}}} \mathbf {s} = \frac{3}{M^2-1} A^2 + c^2 \alpha$ into (43), we obtain

$$\begin{aligned} \mathrm {SINR} = \frac{{\mathcal {D}} - c^2\alpha }{\alpha - 2 c \alpha + c^2 \alpha + \sigma ^2 } \end{aligned}$$

(44)

By direct differentiation of the (44) with respect to c and root selection, we obtain $c^{\mathrm {opt}} = \frac{\alpha + \sigma _n^2 + {\mathcal {D}} - \sqrt{(\alpha + \sigma _n^2 + {\mathcal {D}} )^2 - 4\alpha {\mathcal {D}} } }{2 \alpha }$ in (14). With optimal transform parameter $c^{\mathrm {opt}}$, the optimal amplitude $A^{\mathrm {opt}}$ and the maximum SINR can be easily calculated.

The SINR in (44) is a monotonically decreasing function of $\alpha \ge 0$. Therefore, the optimal carrier $\mathbf {s}$, which minimizes $\alpha \triangleq \mathbf {s}^T \mathbf {R}_{\mathrm { {x}}} \mathbf {s}$, is the eigenvector of $\mathbf {R}_{\mathrm {x}}$ with minimum eigenvalue

$$\begin{aligned} \mathbf {s}^{\mathrm {opt}} = \mathbf {q}_L . \end{aligned}$$

(45)

With this optimal carrier, the identity of match filter and maximum SINR filter has be proved in Proposition 3 in [12]. $\square$